Turbulent rotating convection

Abstract

Many flows in nature and technology are driven by buoyant convection and subsequently modulated by rotation. Typical and highly relevant examples are found in the large-scale geophysical flows in the atmosphere and the oceans on our Earth. A simple yet relevant model is found in rotating Rayleigh–B´enard convection: a fluid layer enclosed vertically between parallel rotating walls is heated from below and cooled from above. The aforementioned flows typically have no lateral confinement (atmosphere), or a very large horizontal extent compared to depth (ocean basins). To experimentally investigate such flows some lateral confinement must be introduced. In numerical simulations another opportunity arises to cope with the lateral directions: the application of periodicity on the lateral boundaries of the computational domain approximates a horizontally infinite fluid layer. In this thesis four investigations of turbulent rotating convection are combined. In the first part turbulent rotating convection is investigated experimentally. Local in situ velocity measurements are performed using stereoscopic particle image velocimetry (SPIV) in a cylindrical convection cell of diameter-to-height aspect ratio one, placed on top of a rotating table. SPIV is a nonintrusive method that measures the three components of velocity at many positions in a two-dimensional cross-section of the fluid. Experiments at various rotation rates showed that the organisation of the flow into coherent structures is strongly dependent on rotation. For small rotation rates the domain-filling large-scale circulation, well-known from non-rotating convection, is the dominant feature. At larger rotation rates an irregular, unsteady array of vortical plumes is found. The turbulence intensity is reduced by rotation, and the vertical inhomogeneity increases. Numerical simulations of turbulent rotating convection in a cylinder cover the second part of the thesis, to compare with and expand on the experimental results. The Navier–Stokes and heat equations are written in cylindrical coordinates and discretised using second-order accurate finite-difference approximations for the derivatives. The cylinder axis is treated separately to avoid the singularities that arise there in the formulation in cylindrical coordinates. The simulations confirm the findings from the experiments. Additionally, it is found that, despite the reduc- 185 tion of turbulence intensity, the convective heat transfer through the fluid layer is enhanced in a certain range of rotation rates. The vortical plumes are responsible for nearly all vertical transport of fluid and heat. They possess an efficient means for entraining boundary-layer fluid and transporting it towards the vertically opposite side: Ekman pumping. Ekman pumping enhances the heat transfer. At the highest rotation rates, however, the inhibition of velocity by rotation dominates and the heat flux decreases abruptly. Another numerical approach is employed in the third part. The computational domain is rectilinear. Periodic boundary conditions are applied in the horizontal directions and solid boundaries vertically. The discretisation employs formulations of the discrete derivatives that preserve the (skew-)symmetry of the original differential operators, which, combined with a Richardson extrapolation, attains fourth-order accuracy. The (skew-)symmetric formulation ensures mass and momentum conservation, as well as stability. With a variation of the rotation rate between simulations similar trends as in the other two parts are found. Evaluation of the turbulent kinetic energy budget revealed the Ekman pumping contribution in the turbulent transport term. The fourth part is a theoretical investigation. Under rapid rotation the vortical plumes that are formed are nearly symmetric in the mid-plane. A prescription of these symmetries in the governing equations, with the vertical boundary conditions as before, has led to a vortex solution which possesses many of the features of the vortical plumes found in the numerical simulations. This vortex solution has also been used to derive a relevant upper bound on the heat transfer. The vortex solution can be used as a model for the vortical plumes of strongly rotating convection. The thesis concludes with a discussion of the structure function scaling in turbulent (non-rotating) convection. There is an ongoing debate in the literature on whether turbulent convection possesses a range of scales for which the so-called Bolgiano–Obukhov scaling is valid. This scaling occurs when the temperature acts as an active scalar, rather than the Kolmogorov–Obukhov–Corrsin scaling when temperature is passive. It is shown that the length scale that separates the two ranges, the Bolgiano length, can vary considerably in magnitude depending on the measurement position within the flow domain. It is much larger than the oft-used a priori estimate based on the domain-integrated dissipation rates. Taking into account the local Bolgiano scale, the Bolgiano–Obukhov scaling range has been found both in the experiment and in the numerical simulations. The results from the various investigations presented in this thesis elucidate the effects of rotation on the statistics and the formation of coherent structures in convective turbulence. It has provided valuable input for the modelling of convective flow under rotation in, e.g., climate models.